Intro to DFT+U

Slides created by Heather Kulik intended for educational use only. Visit http://www.stanford.edu/~hkulik for more info.

Recent developments in
Hubbard-augmented DFT
Heather Kulik
02/03/12


Nicola Marzari
MIT/EPFL Quantum-ESPRESSO

Matteo Cococcioni
U Minnesota

http://www.quantum-espresso.org
Open source plane-wave, pseudopotential code
Other codes with similar implementations:
VASP, ONETEP, Qbox, others?
Coming soon: TeraChem, GPAW?


http://www.stanford.edu/~hkulik


Density functional theory
Exact…in theory
One-to-one mapping of many-body interacting system
onto a non-interacting one.
Quantum mechanis becomes computationally
tractable.


Exact…in theory
tractable.

Approximations in practice


Exact…in theory
tractable.

Approximations in practice
Charge transfer (short or long range)
Electron delocalization
Wrong dissociations
…all some form of self-interaction error.


Electronic structure methods
A wavefunction worldview
A density worldview
Hartree-Fock/MCSCF
higher derivatives of the density Perturbative theories + RAS/CAS/etc.
adding in Hartree-Fock exchange Coupled cluster methods
parameterizing until the (Some approximation to) Full CI
end of time

A “sophisticated” condensed matter
electronic structure worldview
Density matrix renormalization group
Dynamical mean field theory
GW approximation
Quantum Monte Carlo


But I just want results…
My (slightly different)
density worldview
Physics-based, parameter free
methods to alleviate self-
interaction

For 1-1000 atoms (or more with GPUs), approaches that
balance accuracy with computational efficiency.


DFT+U


DFT+U
DFT+U+V


DFT+U
DFT+U+V
DFT+U(R)


DFT+U
DFT+U+V
DFT+U(R)
in practice


Basic Hubbard model Hamiltonian

Conductor to
insulator transition



Conductor to

DFT conductors to
DFT+U insulators



Conductor to

DFT conductors to E
DFT+U insulators

DFT
conductors



Conductor to

DFT conductors to E E
DFT+U insulators

DFT DFT+U
conductors



Conductor to

DFT conductors to E E
DFT+U insulators

DFT DFT+U
conductors  insulators
V.I. Anisimov, J. Zaanen and O.K. Andersen. Phys. Rev. B, (1991).
M. Cococcioni and S. de Gironcoli. Phys. Rev. B, (2005).


DFT+U for molecules
UGE Perera, HJK et al
Phys. Rev. Lett. (2010).

HJK et al J. Am. Chem.
Soc. (2009).
1.0
_
_
_ _ MRCI
6 4

_ DFT+U+
FeOH +CH3
Relative Energy (eV)

0.0

_ _
_
-1.0
_
_ _
_ _
_
_
-2.0
_
_
HJK et al Phys. Rev. Lett. (2006).
-3.0 HJK et al Phys. Rev. Lett. (2006).
HJK et al/CH Chem. Phys. (2008). Fe /CH OH
FeO J. 1 TS1 2 TS2 +
3
4
+
3
HJK et al Fuel Cell Science (2010).


Physical meaning of DFT+U
Energy of an atom
Energy

N-1 N N+1
# of Electrons

J.P. Perdew, R.G. Parr, M. Levy, and J. L. Balduz, Jr. Phys. Rev. Lett. (1982).
M. Cococcioni and S. de Gironcoli. PRB, 71: 2005.


Energy of an atom
exact
Energy

N-1 N N+1
# of Electrons



Energy of an atom
exact
LDA/GGA
Energy

N-1 N N+1
# of Electrons



Energy of an atom
exact
LDA/GGA
Energy

+U

N-1 N N+1
# of Electrons



Energy of an atom
The “+U” contribution to standard DFT:
exact
LDA+U
Energy

+U

U is the extent of curvature: we
calculate this uniquely for each system.

N-1 N N+1
# of Electrons



Choosing occupations
1) Select the localized manifold or manifolds for each
atom “site”

2) Choose the projections

Results in this talk: Other options:
Wannier/Boys functions
Population schemes


Linear response U
U is the curvature: We calculate it from linear response:

In lieu of constrained occupations

n’
6 + MX
Converged response
(from an SCF calculation)

n

Bare response due to
rigid potential shift on
localized manifold


U is a system-dependent property
A property that should be calculated
6 + MX
MX U (eV)
FeO+ 5.50 Electron configuration
Covalency/ionicity

Less covalent
FeN 4.38 Spin states/charge states
MnO 3.41 Element identity
Coordination numbers
CrO- 2.85
CrF 2.00
Isoelectronic
Series
HJK and N. Marzari, J. Chem. Phys. (2010).


A self-consistent U
Calculate U self-consistently
Most key for when
on the DFT+U system:
DFT and DFT+U
ground states differ

HJK et al., Phys. Rev. Lett. (2006).


DFT+U+V


Extending the Hubbard model


Extending the Hubbard model

J I K

VIJ UII VIK

V favors intersite interactions

J. Hubbard Proc. R. Soc. A 285 (1965). V. I. Anisimov, I. S. Elfimov, N. Hamada, and
J. Hubbard Proc. R. Soc. A 296 (1967). K. Terakura Phys. Rev. B 54 (1996).


Functional form
Extended Hubbard Model

Campo and Cococcioni, J. Phys. Cond. Matt. (2010).


Functional form
Extended Hubbard Model Generalized FLL double counting



Generalized occupations

m and m’ defined by interacting manifolds

nII nIJ
Connection to atomic projections
is clear. Wannier basis less so
nJI nJJ
(already bond-centered?)

Block diagonals: on-site
standard occupations.


What happens to states

nII nIJ

nJI nJJ
Internal competition



Standard U: Favors integer occupations in
block diagonals, weak off-site blocks. nII nIJ

nJI nJJ



Standard U: Favors integer occupations in
block diagonals, weak off-site blocks. nII nIJ
New V term: strong intersite occupations
in off diagonal.
nJI nJJ


MO2 bent  linear
Experiments:

180

100
Can theory predict transition? E

Gong, Chem. Rev. 2009 and references therein. q


MnO2: Single or double well?
0.8
rMn-O=1.55Å rMn-O=1.70Å rMn-O=1.85Å
0.7
Relative energy (eV)

0.6
U=6
0.5 U=4
U=0
0.4

0.3

0.2

0.1

0.0
110 130 150 170 110 130 150 170 110 130 150 170
O-Mn-O Angle (o) O-Mn-O Angle (o) O-Mn-O Angle (o)

r


MnO2 hybridization

r


O-M-O Structures
Angles Bonds

DFT +U +U+V
MnO2 1.61 1.70 1.59
2
FeO2 1.59 1.67 1.58
CoO2 1.55 1.63 1.56

2
DFT
+U +U|r0: angle from
+U|r0 M-O bond fixed
+U+V to DFT value.
2 Expt.

HJK and N. Marzari, J. Chem. Phys. 134, 094103 (2011).


FeO2 Splitting and Angle

Expt GS

GS
U=
0V=  
0
U=
5V=  
0
U=
5V=  
2

+U +V


Solid state applications
LDA+DMFT+V for VO2 Monoclinic
M1
Cheaper than cluster DMFT but yields
similar results.

Magnetic susceptibilities

A. S. Belozerov, et al. PRB (2012).


Solid state applications
NiO
Cubic rock-salt
structure

Si and
GaAs



DFT+U(R)


Inspiration for a variable U

Errors for 22 MX (X=H,C,N,O,F)
0.40 GGA
0.35 GGA+U
0.30

0.25
Error

0.20

0.15

0.10

0.05

0.00
re e De E
(cm-
(Åx10) (eV) (eV)
1/100

)
HJK and N. Marzari. J. Chem. Phys. (2010).
HJK and N. Marzari, J. Chem. Phys. (2011).



Errors for 22 MX (X=H,C,N,O,F)
0.40 GGA
0.35 GGA+U
0.30

0.25
Error

0.20

0.15

0.10

0.05

0.00
re e De E
(cm-
(Åx10) (eV) (eV)
1/100 In DFT+U, we average U
) over all points. Works
HJK and N. Marzari, J. Chem. Phys. (2011). well most of the time!


Electronic structure in
Errors for 22 MX (X=H,C,N,O,F) differing bonding regimes
0.40 GGA
0.35 GGA+U
0.30

0.25
Error

0.20

0.15

0.10

0.05

0.00
re e De E
(cm-
(Åx10) (eV) (eV)
HJK and N. Marzari, J. Chem. Phys. (2011). well most of the time!


Electronic structure in
Errors for 22 MX (X=H,C,N,O,F) differing bonding regimes
0.40 GGA
0.35 GGA+U
0.30

0.25
Error

0.20

0.15

0.10

0.05

0.00
re e De E
(cm-
(Åx10) (eV) (eV)
DFT+U(R), changes
HJK and N. Marzari. J. Chem. Phys. (2010). in U incorporated
HJK and N. Marzari, J. Chem. Phys. (2011). well most ofkey cases.
directly for the time!


Even better with DFT+U(R)



4

2

0
dE/dR (eV/Å)

-2 Interpolated
-4

-6

-8
DFT+U Forces
-10
1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4
Fe-O Distance (Å)



4 4.0
2
CC value

0 3.0
dE/dR (eV/Å)

-2 Interpolated
-4 2.0

-6
1.0
-8
DFT+U Forces 0 U 6
-10
0.0
1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4
Fe-O Distance (Å) Fe-O Distance (Å)



4 4.0
2
CC value

0 3.0
dE/dR (eV/Å)

-2 Interpolated
-4 2.0

-6
1.0
-8
DFT+U Forces 0 U 6
-10
0.0
1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4
Fe-O Distance (Å) Fe-O Distance (Å)

In practice, interpolate over forces or interpolate over
energies with a common physical reference.


U variation from occupations



Component of
forces gradient



Component of From linear
forces gradient response



6
6
U
Actualfwd.diff.
U0
5
Predicted
Hubbard U (eV)
U (eV) 4

3

2

1 4 FeO+: U vs. R
00
1.6
1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6
2.6
R (Å)
Internuclear Separation (Å)


Predicting U variation from forces


Predicting U variation from forces

Exiting linear regime for
derivatives of forces is a
numerical challenge.


Numerical noise in practice
Predicted U trends for 4 FeO+



In principle, the
force-based
approach is more
exact. In practice, it
suffers from a
greater degree of
numerical noise.


When U(R) matters
A metric: when is U ½ of lin.resp.
U?


When U(R) matters
U?

Molecule U dU/dR rU½
2 + CoC 4.8 -4.0 0.6
2 - CrN 4.3 -2.3 0.9
+ FeO+ 6.3 -5.0 0.6
5 + MnF 2.4 -4.8 0.2
6 + CrF 2.0 -0.1 9.0


When U(R) matters
U?

Molecule U dU/dR rU½ Including more variables
2 + CoC 4.8 -4.0 0.6
2 - CrN 4.3 -2.3 0.9
+ FeO+ 6.3 -5.0 0.6
5 + MnF 2.4 -4.8 0.2
6 + CrF 2.0 -0.1 9.0


When U(R) matters
U?

Molecule U dU/dR rU½ Including more variables
2 + CoC 4.8 -4.0 0.6
2 - CrN 4.3 -2.3 0.9
+ FeO+ 6.3 -5.0 0.6
5 + MnF 2.4 -4.8 0.2
6 +
Some matter
CrF 2.0 -0.1 9.0
more than
others


Ordering multiple U(R) surfaces

Expt.


Ordering multiple U(R) surfaces

Aligned at the effective
united atom limit

Expt.


DFT+U(R) Improvements
1) Binding curves: 2) Reaction coordinates:
Errors on worst case subset H2 on FeO+
from MX DFT+U
re (Å)
CC value
De(eV
)

e (cm-1)

3) Work in progress:
Molecular adsorbates on TM
surfaces. Preliminary evidence:
U(R) improves binding energies.


in practice


Numerical instabilities
Example:

Full manifolds or
integer occupations

Unperturbed or
rigid occupations


Numerical instabilities


Projection dependence



DFT: significant PSP dependence



DFT: significant PSP dependence

+U: Different Us, less PSP dependence


Multiple manifolds
Strong hybridization between 3d and 4s in TM hydrides

dd ds
sd ss

U3d=( -1 -1)
0 - dd

U4s=( -1- -1)
0 ss

In the solid state: Ce 4f/5d/6s, MOFs?


Angle dependence of n and
5
4 bent
3
2
1
0
4.5 5.5 6.5 7.5
5
4 linea
3
2
r
1
0
4.5 5.5 6.5 7.5



5
4 bent
3
2
1
0
4.5 5.5 6.5 7.5
5
4 linear
3
2
1
0
4.5 5.5 6.5 7.5


A renormalized U
Redefining response
functions:

An equivalent U along a
coordinate:

All dependence of U on O-Mn-O angle is
from filling/emptying states!


Conclusions

For transition metals and materials with
localized electrons:
DFT+U-works well in most cases
DFT+U+V-a balance of
localization/delocalization, more general cases
like semiconductors
DFT+U(R)-bond breaking for chemical
applications
In practice, things don’t always go according to
plan (method is still not a black box).

Intro to DFT+U

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Intro to DFT+U

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